Posts Tagged ‘agincourt’

A couple of submitted solutions on the theme of isolating small pieces in tilings of polyominoes of mixed sizes:

George Sicherman solved problem #44. (Tile a certain torus with the 1–5-ominoes such that the 1–4-ominoes do not touch each other. Finding a solution where none of the smaller pieces even meet at corners was optional, but well appreciated!) This solution contains four “crossroads”, or points where four polyominoes meet. These are sometimes considered an aesthetic flaw in a polyomino tiling, and whether or not you agree with that, finding solutions without them tends to be good for an extra challenge.

Problem #44a: find a solution for problem #44 without crossroads.

Jaap Scherphuis analyzed this challenge that was included with my Agincourt puzzle: find a tiling of an 8×8 square where none of the dominoes and trominoes touch each other or the outer edge. He found that there were 32 solutions. In just one of these, none of the smaller pieces meet at a corner:

I don’t feel that I acknowledge other puzzle creators enough in this space, so I want to give a shout out to Kadon for their Mini-Iamond Ring puzzle, which contains all of the 2–5-iamonds, and includes as a challenge isolating all of the different sizes of pieces:

It lately occurred to me that there are concepts that I use (and see used by others) in creating variations on polyform puzzles that I haven’t seen explained very thoroughly, and it might be helpful if I used this space for just that purpose.

Some polyomino puzzles using symmetry variations

The first of these is the use of different kinds of symmetry in defining the set of pieces used in a puzzle. (I touched on this in my post on rectangular-cell pentominoes.) Normally, all combinations of rotations, translations, and reflections of a polyomino in a grid are considered to be equivalent. Leaving aside translations for the moment, the possible rotations and reflections of a polyomino are equivalent to the group of symmetries of a square. We can find variations on polyominoes by restricting the allowed symmetries to subgroups of that group. For example, the one-sided polyominoes are the result of allowing only rotations, not reflections. Rhombic cell pentominoes (which Kadon sells) allow 180° rotations, plus diagonal reflections. My Agincourt puzzle allows only reflections over vertical axes, assuming that the arrows are pointing vertically. Notice that it doesn’t matter which direction the arrows point as long as they point in the same direction; this suggests that what we are interested in isn’t symmetry subgroups per se, but classes of subgroups where two subgroups that are related to each other by symmetries of the square are equivalent.

What are all of the possible variations with different allowed transformations? We can generate a representative subgroup of every class by using some combination of reflection over a particular axis parallel to the grid, a particular diagonal axis, and 90° and 180° rotations. Here’s a chart of the symmetry variations this produces.

Polyomino Type

Reflection

Rotation

# of Symmetries

Free

Either

90°

8

Parallel (a.k.a. Rectangular)

y axis

180°

4

Diagonal (a.k.a. Rhombic)

x=y

180°

4

One-sided

None

90°

4

Oriented Parallel

y axis

None

2

Oriented Diagonal

x=y

None

2

Polar One-sided

None

180°

2

Fixed

None

None

1

I chose the above terminology for the types (after keeping “free”, “one-sided”, and “fixed” as established terms) in order to build in some helpful mnemonics. The types with four symmetries have short names. The types with two symmetries have longer names based on the names of the types whose symmetry groups their symmetries are subgroups of. The odd duck here is “polar one-sided”, which is a subgroup of all of the larger symmetry groups, but putting “one-sided” in its name makes the types with two symmetries nicely echo the names of those with four.

Here’s a chart of the number of polyominoes of each type for a given size:

(The odd entries for the polar one-sided polyominoes track those for the oriented parallel polyominoes exactly for several terms, before eventually diverging. There are 4998 9-ominoes for both, and 67792 polar one-sided, and 67791 oriented parallel 11-ominoes. It seems unlikely that this is a coincidence. Does anyone know why this occurs?)

These types can be realized geometrically by replacing square cells in a planar tiling with cells with the appropriate symmetry. Another way they can be realized is by keeping the cells square and marking them with a figure with the appropriate symmetry. This is essentially what I did by cutting arrow shaped holes in the Agincourt pieces. The latter method allows the possibility of mixing different symmetry types in the same tiling. I don’t believe I’ve seen such a problem before, so let me be the first to fill what may be a much needed gap:

Problem #28: Tile a 6×6 square with the oriented parallel, oriented diagonal, and polar one-sided trominoes. No tromino should touch another of the same type.

With these symmetry subgroup based polyform variations in mind, any type of polyform on a square grid can be transformed into an entire family of polyforms. In particular, polysticks would reward exploration in this light, which does not seem to have occurred yet. A similar analysis to the one above can be made for symmetry based variations of polyiamonds and polyhexes. Bringing translation symmetry subgroups into the picture leads to things like checkered polyominoes. I may get to these in later posts; this one was getting long enough that I needed to wrap it up.

I should note that Peter Esser’s pages on polyforms cover these variations, and that his polyomino solver program can work with any of the 8 symmetry types (but not with mixed types.) (It is, sadly, a Windows binary, but I’ve been able to make it work under Wine on Linux.)

Agincourt is one of the lasercut acrylic puzzles which I’m selling through the store. It’s the set of all of the ways to make 2-, 3-, and 4-ominoes with arrow shaped holes in each square pointing in the same direction. The symmetry of the arrows means that you can flip over pieces without changing the arrow directions, but you can’t rotate them. Most of the puzzles I have designed for the set ask for the solver to make all pieces point the same way, but the arrows naturally suggest a scoring system to handicap the puzzle for different levels of solvers — just count the number of pieces you had to put in the wrong direction, and try to improve on your score.

Here’s a solution to the puzzle that literally comes out of the box. (The puzzle comes in the box with 4 layers of pieces in 4 × 4 squares.)

I have had a lifelong interest in recreational mathematics, especially polyomino problems. I’ve produced some puzzles in laser-cut plastic, which I sell via this very site. I’ve also dabbled in writing interactive fiction.